All Questions
Tagged with chow-groups ag.algebraic-geometry
100 questions
0
votes
0
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79
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Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
1
vote
0
answers
97
views
Weil restriction of cycles and norm algebra
This question is on a concrete descrption of weil restricton of an affine algebra.
Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
- 63.9k
1
vote
0
answers
80
views
Computing Chow groups of affine, simplicial toric varieties
Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
- 639
4
votes
0
answers
167
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Is the group of homologically trivial cycles in a variety over a finite field torsion?
Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
- 531
1
vote
0
answers
134
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Paper request: Alberto Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians
I am trying to locate a copy of the paper by Alberto Collino titled "Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians".
I can neither download nor purchase ...
- 441
5
votes
0
answers
157
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Motives with compact support, Chow groups and proper pushforward maps
In Motivic cohomology of smooth geometrically cellular varieties (1999), Corollary 3.5, Bruno Kahn proves the following statement. Consider a cellular variety $X$ (i.e. it admits a filtration by ...
- 3,004
4
votes
0
answers
135
views
Specialization map Chow groups preserves algebraic equivalence
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$.
Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers.
In ...
- 984
1
vote
0
answers
89
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Vanishing of chow group of 0-cycles for affine, simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $X$ be an affine, simplicial toric variety over $k$.
If $X$ has dimension one, then it is the affine line over the field $k$, so ...
- 639
1
vote
0
answers
96
views
Chow ring of simplicial toric varieties
Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that ...
- 639
3
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0
answers
152
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Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?
Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
- 984
1
vote
0
answers
137
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Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
1
vote
0
answers
162
views
Bloch's higher Chow group as relative ordinary Chow group
If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of ...
- 219
1
vote
0
answers
136
views
Universal properties for Bloch's higher Chow groups
I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow ...
- 219
2
votes
0
answers
292
views
Tate's conjecture for arithmetic schemes
Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
- 5,901
1
vote
2
answers
279
views
Correspondences acting on cohomology groups $H^*(X)$ & splittings
Let $X$ be a smooth connected proper scheme over field $k$. It is known that
correspondences $\alpha \subset X \times X$ regarded as
objects in Chow groups $\text{CH}^*(X \times X)$
act on cohomology $...
- 40.3k
2
votes
0
answers
60
views
Composition of correspondences pulled back to $\mathrm{CH}_0$
Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition,
$$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...
- 3,730
1
vote
1
answer
452
views
The Ogus conjecture for crystalline cohomology
How is the Ogus conjecture explicitly stated, which is a variant of the Hodge and the Tate conjectures for crystalline cohomology ?
How do we build its class cycle map, and how do we formulate its ...
- 5,407
3
votes
0
answers
248
views
Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?
A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
- 12.9k
3
votes
0
answers
338
views
Pushforward and pullback on the level of Chow varieties
Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic
...
- 5,901
1
vote
0
answers
254
views
A question on the Chow group on stacks
Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...
- 565
0
votes
0
answers
75
views
General fiber and the symmetric product of an ample hypersurface
Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
- 519
1
vote
1
answer
217
views
Meaning of torsion points in a Roitman's theorem
I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
- 35.2k
2
votes
0
answers
125
views
About finite dimensionality of Chow groups of zero cycles
Let $S$ be a connected smooth complex projective surface.
Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$.
Let $Sym^{d,d}(S)=...
- 519
4
votes
0
answers
114
views
Chow group of different reductions of a smooth projective variety
Let $(R,\mathfrak{m},k)$ be a discrete valuation ring with fraction field $K$. Let $X/K$ be a smooth projective variety.
Let $\mathcal{X},\mathcal{X}'$ be smooth projective models of $X$ over $R$. Let ...
CommunityBot
- 1
1
vote
0
answers
102
views
About Definition 2 in Roĭtman's Paper
Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero.
In Definition 2 of Roĭtman's paper ...
- 519
1
vote
0
answers
213
views
Algebraic correspondence as morphisms in Betti cohomology
$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
- 63.9k
1
vote
1
answer
373
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Singularities of Chow varieties
Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...
- 4,111
1
vote
0
answers
77
views
Non-torsion infinitely divisible elements in the Chow group
It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex ...
- 5,901
0
votes
0
answers
145
views
Chow countability argument
I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!
- 519
7
votes
1
answer
474
views
Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
CommunityBot
- 1
0
votes
1
answer
224
views
Notation on a Mumford's paper
I am reading the paper " D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1969) 195 - 204" and I do not understand a notation in bottom of page 197. It ...
CommunityBot
- 1
2
votes
0
answers
64
views
Motivic complexes associated to adequate equivalence relations
Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic ...
- 5,901
0
votes
0
answers
172
views
Pullback of algebraic $K$-theory along the surjection of abelian varieties
Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
- 5,901
2
votes
0
answers
274
views
Generic rank of proper pushforward of the trivial line bundle
Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
- 5,901
2
votes
0
answers
90
views
Homological and rational adequate equivalences for product of curves
There is a variant of Standard Conjecture D for projective varieties over finite fields. It claims that rational and homological equivalences are equivalent on cycles after tensoring with $\mathbb{Q}$....
- 5,901
5
votes
1
answer
264
views
Moduli of rational equivalent classes of 0-cycles
Let $X$ be a smooth variety over a field $k$ and I'd like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I'm wondering if there is any reasonable ...
- 2,327
0
votes
0
answers
129
views
Injectivity of pushforward of rational Chow groups
I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...
- 5,901
2
votes
1
answer
409
views
Invariance of Chow groups of projective bundles under automorphisms of bundles
Suppose $X$ is a smooth scheme, $E=O_X^{\oplus n}$ and $\varphi\in SL_n(E)$, i.e. $\varphi$ has trivial determinant and is an isomorphism. Is the morphism
$$\mathbb{P}(\varphi)^*:CH^{\bullet}(\mathbb{...
- 342
5
votes
0
answers
191
views
Applications of Chow rings of classifying spaces in algebraic geometry
For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of
Totaro, The Chow ring of a classifying space
has been computed in many cases. Is there any interesting ...
- 935
3
votes
0
answers
280
views
Descent and Chow groups
One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have
$$[X,K(\mathbb{Z}(n),2n)]\...
- 63.9k
2
votes
0
answers
239
views
Localization of Chow groups and flat base change
For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$Ch^i(Y)\rightarrow Ch^i(X).$$
A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
- 4,418
9
votes
0
answers
278
views
Which field extensions do not affect Chow groups?
Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...
- 16.9k
4
votes
2
answers
459
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 6,734
3
votes
0
answers
914
views
Relations between rational algebraic K-theory and Chow groups
A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
- 6,622
2
votes
1
answer
585
views
Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $
Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph:
For any scheme of finite type over a ground field ...
- 5,429
6
votes
0
answers
203
views
Chow ring of $E_7$ varieties
Consider a split algebraic group $G$ of type $E_7$ over a field of characteristic zero.
It is known that some subgroups $P_i$ of $G$, which are called parabolic, have the property that the object $G/...
- 6,881
11
votes
1
answer
737
views
Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$
What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?
I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be ...
- 2,821
4
votes
1
answer
255
views
Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m $
Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{...
- 153
1
vote
0
answers
116
views
Numerical and rational equivalences on intersection of divisors
Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
- 5,901
2
votes
1
answer
542
views
Question regarding Chow group of a blow-up
Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \...
- 149k